rashid mehmood - hec
TRANSCRIPT
1
Non Orthogonal Stagnation Point Flows with
Rheological Characteristics
By
Rashid Mehmood Department of Mathematics
Quaid-i-Azam University
Islamabad, Pakistan
2015
2
Non Orthogonal Stagnation Point Flows with
Rheological Characteristics
By
Rashid Mehmood Supervised By
Dr. Sohail Nadeem
Department of Mathematics
Quaid-i-Azam University
Islamabad, Pakistan
2015
3
Non Orthogonal Stagnation Point Flows with
Rheological Characteristics
By
Rashid Mehmood Dissertation Submitted in the Partial Fulfillment of the Requirements
for the Degree of
DOCTOR OF PHILOSOPHY
IN
MATHEMATICS
Supervised By
Dr. Sohail Nadeem
Department of Mathematics
Quaid-i-Azam University
Islamabad, Pakistan
2015
4
Contents
Nomenclature 7
1 Introduction 9
2 Optimized analytical Solution for oblique flow of a Casson-nano fluid with convective
boundary conditions
2.1 Introduction……………………………………………………………………………..17
2.2 Mathematical formulation………………………………………………………………18
2.3 Solution by Optimal Homotopy analysis method……………………………………...23
2.4 Optimal convergence control parameters…………………………………………….....28
2.5 Results and discussion…………………………………………………………………..30
2.6 Conclusions…………………………………………………………………………….44
3 Combined effects of magnetic field and partial slip on obliquely striking rheological
fluid over a stretching surface
3.1 Introduction………………………………………………………………………….…46
3.2 Mathematical Formulation…………………………………………………………..…47
3.3 Numerical Solution…………………………………………………………………..…50
3.4 Results and discussion…………………………………………………………………..51
3.5 Concluding Remarks………………………………………………………………….....60
5
4 Oblique stagnation flow of Jeffery fluid over a stretching convective surface: Optimal
Solution
4.1 Introduction……………………………………………………………………………..62
4.2 Mathematical formulation………………………………………………………………63
4.3 Solution by Optimal Homotopy analysis method……………………………………...67
4.4 Optimal convergence control parameters………………………………………………68
4.5 Results and discussion………………………………………………………………….70
4.6 Conclusions……………………………………………………………………………..80
5 A comparative study on flow and heat transfer analysis for a non-aligned Jeffery
nanofluid over a stretching surface
5.1 Introduction…………………………………………………………………………....82
5.2 Mathematical Analysis…………………………………………………………………83
5.3 Non-dimensional quantities of interest…………………………………………..….….86
5.4 Method of solution……………………………………………………………………..86
5.4.1 Numerical solution……………………………………………………………………86
5.4.2 Analytical solution by Optimal Ham………………………………………………….87
5.4.3 Optimal convergence control parameters……………………………………………..87
5.5 Results and discussion……………………………………………………………….…91
5.6 Conclusions……………………………………………………………………………...101
6 MHD Second grade oblique flow over a convective surface with soret and dufour effects
6.1 Introduction……………………………………………………………….……………..102
6.2 Mathematical Formulation………………………………………….……………………102
6.3 Numerical Solution……………………………………………………………………...107
6
6.4 Results and discussion………………………………………………………………….107
6.5 Concluding Remarks……………………………………………………………………123
7 Numerical investigation on MHD oblique stagnation point flow of a Walter’s B type nano
fluid over a convective surface
7.1 Introduction……………………………………………………………………………..124
7.2 Mathematical Formulation……………………………………………………………...125
7.3 Numerical Solution……………………………………………………………………...129
7.4 Graphical Results and Analysis…………………………………………………….…...132
7.5 Conclusion……………………………………………………………………………....150
8 Effect of internal heat generation/absorption on an obliquely striking Maxwell fluid past
a convective surface
8.1 Introduction……………………………………………………………………………..151
8.2 Problem Formulation……………………………………………………………………151
8.3 Non-dimensional quantity of interest…………………………………..……………….155
8.4 Numerical solution………………………………………………………………………155
8.5 Analysis of Graphical Results ……………………………………………………….…159
8.6 Concluding Remarks……………………………………………………………………167
7
Nomenclature
,
,
( ) ( )
8
⁄
( )
( )
( )
( )
( )
( )
9
Chapter 1
Introduction
Stagnation point flows are universal as they certainly appear to be composite flow fields. Such
flows are either stagnated by a compact wall or by means of a free stagnation point in the fluid
domain. They can be viscous or in viscid, steady or unsteady, 2D or 3D and orthogonal or non-
orthogonal. The typical problems of 2D and 3D stagnation point flow are manipulated by
Hiemenz [1] and Howarth [2] respectively. The 2D stagnation point flow is among the most
widely studied problems in the field of fluid dynamics. Such flows arise when a fluid slants the
boundary of a surface, for example, on an aircraft wing or on an oscillating cylinder immersed in
fluid. These flows have a stagnation-point existent in the fluid and their streamlines locally look
like those nearby a saddle point. Blood flow at a junction within an artery can be considered as
an example of such flows. Stuart [3] devised the non-orthogonal stagnation point flow over a flat
surface, an idea which was later on invigorated by Tamada [4] and Dorrepaal [5]. Stagnation
point flow is a problem of speculative and practical interest in terms of heat transfer analysis.
The importance of stagnation point heat transfer in problems such as atmospheric re-entry and
other esoteric hypersonic flows make reckoning the heat transfer, a problem of concrete
engineering concern. Chiam [6] discussed the stagnated flow towards a stretched surface. Gupta
et al [7] investigated heat transfer in stagnation point flow towards stretching sheet. Mahapatra et
al [8] and Ishak et al [9] investigated hydro magnetic stagnation-point flow towards a stretching
and shrinking surface respectively. Reza et al [10] examined the steady oblique stagnation point
flow passed a stretching surface. Lok et al [11] discussed non-orthogonal stagnation point flow
of a micropolar fluid. Gupta et al [12] inferred the viscoelastic fluid flow for the case of a slanted
10
stagnation-point and later on highlighted some useful aspects of non-orthogonal stagnation point
flow over a stretched surface [13]. Pop et al [14] examined the heat transfer of a non-orthogonal
stagnation point flow in a non-Newtonian fluid. Similarly, Bagheri et al [15] analytically
discussed the problem of non-orthogonal stagnation point flow over a flat sheet. Recently,
rheological fluids are much appreciated for their importance in industrial applications such as
extrusion of plastic sheets, fabrication of adhesives and in coating of inelastic substrates. It
appliestosubstanceswhichhaveacomplex structure includingmuds, sludge’s, suspensions,
emulsions andotherbiologicalmaterials. Rheological fluids are more complex in nature so
their non-dimensional properties are not easy to understand. Walters [16] discussed the second-
order effects in elasticity. Among the class of several other non-Newtonian fluids one important
fluid model is the Maxwell fluid model [17]. This model has the beauty that it relates the
relaxation time with the stress which proves to be very valuable in defining the response of
geological and many other polymeric fluids [18]. Nakamura et al [19] numerically studied the
flow through an axisymmetric stenosis considering non-Newtonian fluid. Similarly Abel et al
[20] investigated hydromagnetic viscoelastic fluid with heat source and variable thermal
conductivity. Recently, Ahmad and Asghar [21] conducted a valuable study on flow of a MHD
second grade fluid over an arbitrary stretching sheet. Liao et al [22] analysed the non-Newtonian
flow over a stretching surface. Xu et al [23] analytically discussed unsteady MHD flow over a
stretching surface. Nazar et al [24] numerically deliberated three-dimensional viscoelastic flow
past a stretching surface. Prasad et al [25] discussed MHD viscoelastic fluid with variable
viscosity.
In certain engineering processes like paper production, glass blowing, crystal growing and metal
spinning, heat transfer at the stretching surface is dynamic since it is closely related with the
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quality of final product. Therefore, it is exceptionally valuable if we can control the heat transfer
rate at the surface. A nano fluid comes up with the solution to meet such debits. Choi [26]
devised the notable concept of nanofluids. In fact the notion of nanofluids have become a topic
of global interest in the last few decades as they provide an effective way of improving heat
transfer characteristics of fluids. It is expected that nanofluids have higher thermal conductivity
than that of common fluids. Heat transfer fluids like water, Oil and Glycols are extensively used
in industrial and civil applications such as hot metal shears, fire-resistance, air conditioning,
electric cooling etc but they possess low thermal conductivity. In order to overcome this
deficiency, thermally conductive Nano meter sized particles (1-100 nm) are suspended in fluid to
accelerate heat transfer at the wall. Putra et al [27] studied the natural convection of nano fluids.
Li et al [28] highlighted some useful features of nano fluids. Buongiorno [29] presented a
mathematical model to describe the flow characteristics of nano fluids. Nield [30] discussed the
flow of a nano fluid past a vertical surface with natural convection. Ishak et al [31] inspected the
stagnation-point flow over a stretched sheet in a nano fluid. Hydro-magnetic fluids are given a
lot of respect in the last few decades as they can be quite useful in certain flow problems like in
refrigeration coils, pumps, power generators, filament cooling and in the purification process of
liquid metals. Anderson [32] discussed the MHD flow of a viscoelastic fluid past a stretching
surface. Pop et al [33] re-investigated the similar kind of problem for the case of stagnation point
flow. Ellahi et al [34] presented the analytical solutions of MHD third-grade fluid flow with
variable viscosity. Nazar et al [35] inspected the influence of induced magnetic field on flow
over a stretching surface. It is a well-known fact that no-slip condition for various rheological
fluids is not good enough, as certain polymer melts exhibits wall slip. Such kind of flows have
their own applications like in synthetic heart valves refining and flows in interior cavities. Navier
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[36] suggested that this wall slip can be related with the shear stress of the fluid. Later on, a
number of researchers have invoked this astonishing concept in their studies. Anderson [37]
discussed the flow over a stretching surface with slip factor. Wang [38] perceived the partial slip
flow on a stretching surface and further extended the work for stagnation point flows with slip
[39]. Fotini et al [40] examined the stagnation point flow for Walter’s B fluid in the presence of
surface slip. In another article, Ariel [41] addressed stagnation point slip flow assuming non-
Newtonian fluid. Turkyilmazoglu [42] presented multiple solutions of MHD viscoelastic slip
flow over a stretching sheet. Fluid flows over convective surface are beneficial precisely in
processes like filtration, petroleum recovery, thermal energy storage and nuclear plants etc. Aziz
[43] examined the thermal boundary layer on a convective flat plate. Makinde [44] deliberated
hydro magnetic flow over a convective vertical plate. Similarly Ishak [45] discussed flow over a
permeable convective surface with heat transfer. In another article, Olanrewaju et al [46]
interpreted flow over a vertically stretching surface with buoyancy effects and convective
boundary condition. Aziz [47] deliberated a study on MHD flow with constant heat flux and later
on inspected the MHD mixed convective flow past a vertical plate inserted in a porous medium
[48]. MHD flow over a convective moving vertical plate has been investigated by Makinde [49].
Rundora et al [50] highlighted the effects of variable viscosity on non-Newtonian fluid with
porous medium and asymmetric convective surface. Soret effect is basically a diffusion flux
which occurs due to temperature gradient in the flow, whereas Dufour effect measures the heat
flux because of a chemical potential gradient. These effects have their own importance in the
areas of geoscience and chemical engineering. Diffusion-thermo effect is extremely vital in
isotropic separation among gases having various molecular weights. Huang et al [51] studied
Hiemenz flow over a porous stretching surface with soret and dufour effects. Prasad et al [52]
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numerically explored Soret and Dufour effects on MHD convective flow over a stretched surface
towards a porous medium. Afify [53] deliberated a useful study to discuss thermo diffusion
effects on a free convective porous medium. Pal et al [54] investigated MHD mixed convective
flow over a stretching sheet with soret dufour and chemical reaction.
Keeping in view all of the above mentioned aspects, stagnation point flows of rheological fluids
over a stretching surface have been presented in which the flow approaches a surface obliquely.
We have applied analytical technique optimal homotopy analysis method (OHAM) for the series
solutions of governing nonlinear equations. Optimal homotopy analysis method (OHAM)
proposed by Liao [55-62] is an extremely effective analytical technique to solve coupled highly
nonlinear ordinary differential equations. In addition numerical schemes known as Spectral
Quasilinearisation method (QLM), Spectral Local Linearization method (SLM) and mid-point
integration scheme along with Richardson’s extrapolation are applied to deal the governing
system of differential equations [63-66].
The chapter wise analysis of this thesis is planned as follow:
Chapter 2 deals with the steady stagnation point flow of a Casson nano fluid with convective
boundary condition. The fluid strikes the wall in an oblique manner. The prevailing nonlinear
partial differential equations of the non-dimensional problem are presented and then converted
into nonlinear ordinary differential equations by means of similar and non-similar variables. The
consequential ordinary differential equations are successfully solved using Optimal Homotopy
analysis method (OHAM). Non-dimensional velocities, temperature, nano particle concentration,
skin friction co-efficient and local heat and mass flux profiles are discussed against emerging
non-dimensional parameters. The contents of this chapter are published in International
14
Journal of Thermal Sciences. (2014):78; 90-100.
Simultaneous effects of partial slip and transversely applied uniform magnetic field on an
oblique stagnation point rheological fluid over a stretched convective surface has been inspected
in chapter 3. The prevailing momentum equations are designed by manipulating casson fluid
model. The obtained differential equations are solved numerically through midpoint integration
scheme together with Richardson’s extrapolation. Numerical values of normal and tangential
components of wall shear stress are tabulated. This chapter is published in Journal of
Magnetism and Magnetic Materials. (2015):378; 457-462.
Chapter 4 is related to the steady oblique stagnation flow of Jeffery fluid over a stretching
surface with convective boundary condition. Optimal homotopy analysis method (OHAM) is
operated to deal the resulting ordinary differential equations. OHAM is found to be extremely
effective analytical technique to obtain convergent series solutions of highly non-linear
differential equations. Graphically, non-dimensional velocities and temperature profile are
expressed. Numerical values of skin friction coefficients and heat flux are computed. This
problem is published in International Journal of Numerical Methods for Heat and fluid
flow. (2015): 25; 454-471.
The main intention of chapter 5 is to present a comparative study on non-orthogonal stagnation
point fluid flow using Jeffery nano fluid as a rheological fluid model. The effects of
thermophoresis and Brownian motion are taken into account. Consequential highly non-linear
system of differential equations is solved numerically through mid-point integration as a basic
15
scheme along with Richardson’s extrapolation as an enhancement scheme and analytical results
are also obtained using optimal homotopy analysis Method (OHAM). Numerical values of local
skin friction coefficients, local heat and mass flux are tabulated numerically as well as
analytically for various non-dimensional parameters emerging in the flow problem. The findings
of this chapter are submitted for publication in Neural Computing and Applications.
Chapter 6 is presented to analyse the soret dufour effects on a steady, 2-D magneto
hydrodynamic (MHD) stagnation point flow of an obliquely striking second grade fluid over a
stretched convective surface. The governing systems of equations are solved numerically using
midpoint integration scheme with Richardson’s extrapolation. The results for normal and
tangential components of velocity as well as temperature and concentration profiles are exhibited
graphically and analysed. Numerical computations are performed to discover the influence of
pertinent non-dimensional parameters on local skin friction co-efficient, heat and mass flux and
are presented in tabulated form. This work is published in European Non-dimensional
Journal plus. (2014)129:182
In Chapter 7, we have numerically investigated the oblique flow of a Walter-B type nano fluid
over a stretching convective surface. Effects of transversely applied magnetic field are also taken
into account. The governing system is presented in the form of highly nonlinear coupled ordinary
differential equations by means of appropriate similarity transformations. These equations have
been solved numerically using the Spectral Quasilinearization Method (QLM) and the Spectral
Local Linearization Method (LLM). Numerical values of skin friction co-efficient, local heat and
16
mass flux are presented. This chapter is published in International Journal of Thermal
Sciences. (2015):95; 162-172.
Chapter 8 studies the influence of internal heat generation/absorption on oblique stagnation
point flow of a Maxwell fluid on a convective surface. By transforming the governing equations
via suitable similarity transformations, numerical solutions are presented using the Spectral
Quasilinearization Method (QLM) and Spectral Local Linearization Method (LLM). The
velocity, temperature and local heat flux rate at the stretching convective surface are presented
through graphs against pertinent flow parameters and analysed. The findings of this chapter are
submitted for publication in Applied Numerical Mathematics.
17
Chapter 2
Optimized analytical solution for oblique flow of a Casson-nano
fluid with convective boundary conditions
2.1 Introduction
This chapter deals with the steady stagnation point flow of a Casson nano fluid with convective
boundary conditions. The fluid strikes the wall in an oblique manner. The governing nonlinear
partial differential equations of the non-dimensional problem are presented and then converted
into nonlinear ordinary differential equations by using similar and non-similar variables. The
resulting ordinary differential equations are successfully solved analytically using Optimal
Homotopy analysis method (OHAM). Non-dimensional velocities, temperature and nano particle
concentration profiles are expressed through graphs. In order to understand the flow behavior at
the stretching convective surface, Numerical values of skin friction co-efficient and local heat
and mass flux are tabulated. Comparison of the present analysis with the previous existing
literature is made and an appreciable agreement in the values is observed for the limiting case.
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2.2 Mathematical formulation
Let us consider the steady 2D oblique flow of Casson-nano fluid over a stretching surface.
Surface is stretched by means of two opposite forces of equal magnitude, as shown in ( )
We further assume that the surface has convective fluid temperature and uniform ambient
temperature here ( )
( ) A non-dimensional description of the problem
The velocity, temperature and concentration functions are defined as
( ) ( ( ) ( )) ( )
( ) ( ) ( )
The governing equations of mass, momentum, energy and concentration are defined as
19
( )
( )
( )
( ) (
) ( )
( )
Here is the velocity, is the fluid density, is the nanoparticle mass density, is the stress
tensor, is the temperature, is the concentration, is the heat flux, is the
thermal conductivity, is the Brownian diffusion coefficient, and is the thermophoresis
coefficient.
The stress tensor of a Casson fluid can be written as , -
( 0
√ 1
[
√ ]
) ( )
where denotes the pressure, is the identity matrix, and is the ( )
deformation rate in component form, is the plastic dynamic viscosity of the non-Newtonian
fluid, is the product of the component of deformation rate with itself, is a critical value of
this product and is yield stress of slurry fluid.
Invoking Eqs ( ) and( ) in Eqs. ( )-( ), we have
( )
(
) ( )
(
) ( )
( )
( ) (
) ( )
20
( )
With the following boundary conditions , -
( )
( )
In above expressions and are the velocity components along the and
respectively, is the kinematic viscosity, is the specific heat of the material, the thermal
diffusivity, is the convective heat transfer coefficient and are positive constants having
dimensions of inverse time.
Introducing the following quantities , -
√
√
√
√
√
( )
Substituting Eq( )into Eqs ( ) -( ), we have
( )
.
/ ( )
.
/ ( )
(
) ( ) ( )
.
/
( )
Introducing the stream function relations
( )
21
Substitution of Eq ( ) into Eqs ( ) -( ) and eliminating the pressure from the resulting
equations using yield
.
/
( )
( ) ( )
(
) ( ) ( )
.
/
( )
The corresponding boundary conditions are defined as
√
( )
( )
where
denotes shear in the stream. We seek solutions of Eqs. ( ) ( ) of the
following form [14]
( ) ( ) ( ) ( ) ( ) ( )
where ( ) and ( ) are normal and tangential components of the flow, and prime indicates
derivative with respect to . Substituting Eq ( ) in Eqs ( ) ( ) we obtain following
system of coupled ordinary differential equations along with the corresponding boundary
conditions
.
/ ( )
.
/ ( )
(
) ( )
( )
22
( ) ( ) ( )
( ) ( ) ( )
( ) ( ( )) ( ) ( ) ( ) ( )
where and represent integral constants. The non-Newtonian (Casson) parameter Prandtl
number the thermophoresis parameter the Brownian motion parameter Biot
number and the Schmidt number are defined as , -
√
( ) ( )
( )
( )
( )
( )
√
( )
Taking the limit in Eq ( ) and using the boundary condition ( )
, we get
(
) Eq. ( ) reveals that ( ) acts as ( )
at where is the
boundary layer displacement constant. Taking in Eq ( ) and making use of the
boundary condition at infinity ( ) we get Thus Eqs. ( ) ( )
become
.
/ (
) ( )
.
/ ( )
(
) ( )
( )
Note that in above equations when we obtain the equations for viscous fluid .
Introducing
( ) ( ) ( )
Using Eq ( ) in Eq ( ) we have
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.
/ ( )
With the boundary conditions
( ) ( ) ( )
Quantities which are of non-dimensional interest are the skin friction coefficient, local heat and
mass diffusion flux rate the surface, which are given in terms of stream function as
(
) (
)
.
/
.
/
( )
In dimensionless form above equations take the following form
(
) ( ( )
( ))
( ) ( ) ( )
The position of attachment of the dividing stream line is determined by the point of zero shear
stress at the wall i.e. and is presented by
( )
( ) ( )
2.3 Solution by Optimal Homotopy analysis method
We now solve these nonlinear boundary value equations by the OHAM, and give an explicit
formula for and First of all, it is obvious that ( ) ( ) ( ) and ( ) can be
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articulated by a set of certain type of exponential base functions , -
* ( ) | + ( )
in the forms
( ) ∑ ∑
( ) ( )
( ) ∑ ∑
( ) ( )
( ) ∑ ∑
( ) ( )
( ) ∑ ∑
( ) ( )
in which
and
are series coefficients. By rule of the solution expressions
along with the boundary conditions ( ) the initial
guesses and of ( ) ( ) ( )and ( )are selected as follow
( )
.
/ ( ( )) ( )
( ) ( )
( ) (
)( ( )) ( )
( ) ( ) ( )
We select the auxiliary linear operators as
( )
( )
( )
( )
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The linear operators in Eqs ( ) ( ) have the following properties
* ( ) ( )+ ( )
* ( ) ( )+ ( )
* ( ) ( )+ ( )
* ( ) ( )+ ( )
where ( ) are the unknown constants.
The zeroth order homotopic deformation equations can be written as
( ) { ( ) ( )} . ( )/ ( )
( ) { ( ) ( )} . ( ) ( )/ ( )
( ) { ( ) ( )} . ( ) ( )/ ( )
( ) { ( ) ( )} . ( ) ( )/ ( )
( ) ( ) ( ) ( ) ( ( )) ( )
( ) ( ) ( ) ( ) ( ) ( )
where , - indicates the embedding parameter and
and
the nonzero auxiliary
parameters. Moreover the nonlinear operators and are prescribed as
. ( )/ .
/
( )
( ) ( )
. ( )
/
.
/
( )
. ( ) ( )/ .
/
( )
( ) ( )
( )
( ) ( )
. ( ) ( )/ ( )
( ( ) ( )
( )
( )
.
( )
/
)
( )
. ( ) ( )/ ( )
( ( )
( )
( )
) ( )
when varies from to we have
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( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) }
( )
By means of Taylor's series
( )
( )
|
( )
( )
|
( )
( )
|
( )
( )
|
( )
The auxiliary converging parameters are chosen in such fashion that the series ( ) converges
when thus we have
( ) ( ) ∑ ( ) ( )
( ) ( ) ∑ ( ) ( )
( ) ( ) ∑ ( ) ( )
( ) ( ) ∑ ( ) ( )
The resulting deformation equations are
* ( ) ( )+
( ) ( )
* ( ) ( )+
( ) ( )
* ( ) ( )+
( ) ( )
* ( ) ( )+
( ) ( )
27
( ) ( )
( ) ( )
( )
( ) ( ( )) ( ) ( ) ( ) ( )
With the following definitions
( ) .
/
∑
∑
(
) ( )
( ) .
/
∑
∑
( )
( )
(∑
∑
∑
) ( )
( )
∑
( )
in which
2
( )
The general solutions of Eqs ( ) ( ) can be written as
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
in which ( )
( ) ( ) and
( ) are the particular solutions of the Eqs( ) ( )
Note that Eqs. ( ) ( ) Can be solved by Mathematica one after the other in the order
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2.4 Optimal convergence-control parameters
We know that our series solutions ( ) ( ) strongly depend upon the auxiliary
parameters
and
which govern the convergence region and the Homotopy solutions
in terms of series . To find out the optimal values of
and
we have utilized the idea of
average residual error given by , -.
∑ *
(∑ ( )
) + ( )
∑ *
(∑ ( ) ∑ ( )
) +
( )
∑ *
(∑ ( ) ∑ ( )
) +
( )
∑ *
(∑ ( ) ∑ ( )
) +
( )
Following Liao , -
( )
where is the total squared residual error, we have considered a case
where
Total average squared residual error is
29
minimized by using Mathematica package which can be found at
http://numericaltank.sjtu.edu.cn/BVPh2_0.htm. We have directly applied the command
Minimize to obtain the corresponding local optimal convergence control parameters.
and are prepared for the case of several optimal convergence control parameters.
shows the minimum value of total averaged squared residual error, while shows the
individual average squared residual error at different orders of approximations using the optimal
values from at . It can be clearly seen that the averaged squared residual errors
and total averaged squared residual errors continuously decreases as we increase the higher order
approximation. Therefore, Optimal Homotopy Analysis Method gives absolute freedom to pick
any set of local convergence control parameters to obtain convergent results.
CPU TIME[S]
: Total Averaged squared residual errors using
CPU TIME[S]
30
20
: Individual Averaged squared residual errors using optimal values at
from .
2.5 Results and discussion
The main objective here is to study the variations of emerging non-dimensional parameters of
nano Casson fluid model. To provide a non-dimensional intuition into our flow governing
equations, ( ) to ( ) are plotted for the velocity field, temperature field,
concentration and the stream lines. ( ) shows the velocity ( )when and different
values of while ( ) shows ( )for
and various values of It is noticed
from ( ) that velocity increases as the stretching ratio is increased and corresponding
boundary layer is found to be inverted. Further, it is quite obvious from Fig. (2.2) that when
1/ ca , the flow has an inverted boundary layer structure. This results in from the fact that
when 1/ ca , the stretching velocity cx of the surface exceeds the stagnation velocity ax of
the external stream. ( ) exhibits that when
velocity ( ) decreases with an
increase in ( ) and ( ) depict the variation of
on ( )for fix and vice versa.
These figures show that initially ( )increases and after ( ) decrease with an increase
in
and ( ( ) ( )) Variation of arising non-dimensional parameters
such as
and on the temperature profile ( )is discussed in ( )
( ) It can be clearly seen from ( ) ( ) that temperature ( ) decreases with
31
higher values of
and it increases with an increase in From ( ) we have found that as
we increase the Prandtl number temperature profiles ( ) and thickness of the thermal
boundary layer tends to decrease which consequently increases the local heat flux.
( ) ( ) show that with the increase of Schmidt number , Brownian
motion thermophoresis parameter and Biot number the temperature profile ( ) also
increases. The effects of non-dimensional parameters
and Schmidt
number on concentration profile ( ) are expressed in ( ) ( ) These figures
suggest that concentration profile ( )increases with an increases in and However, it
decreases for the case of
and The stream line patterns for the oblique flows are shown
in ( ) ( ) The stream line meets the wall at where is the
point of stagnation and zero skin friction. It can be noticed from these figures that the stagnation
point is at the left of the origin for positive values of whereas it is on the right of the
origin for negative values of The shifting of depends upon the magnitude of
– are presented for the numerical values of the skin friction coefficients, local
heat flux and local mass flux for different non-dimensional parameters of interest. From
it can be observed that our computed resulted are in very good agreement with the
available published literature. From it is noticed that magnitude of normal skin friction
coefficient ( ) decreases with an increase in whereas the tangential skin friction
component ( ) tends to increase with increasing
It can be seen from that when
magnitude of ( ) decreases with increasing
for fixed considered. But when
( ) increases with increasing
which is consistent with the fact that there is progressive
thinning of the momentum boundary layer with increase in
is prepared to study the
32
influence of non-Newtonian Casson parameter on skin frictions and local heat and mass flux at
the stretching convective surface. We find out that when is increased both normal and
tangential components of the skin friction also increases, on the other hand it has quite contrary
influence on the local heat and mass flux i.e. it causes to reduce the local heat and mass flux.
is prepared to examine the influence of and for some fix values of
and on the local heat flux respectively. It is obvious from these tables that the magnitude
of ( ) increases for large values of and because heat flows from surface towards the
fluid providing It has been found that heat flux at the surface ( ) decreases for
large values of and It can be seen that effects of the parameters and on the
local mass flux are positive i.e. they causes an increase in the local mass flux. Finally we find out
that Prandtl number thermophoresis parameter and Biot number causes a decrease in
local mass flux when its values are increased.
33
( ) Velocity profile ( ) against when
( ) Velocity profile ( ) against when a .
34
( ) Velocity profile ( ) against when
( ) Velocity profile ( ) against when a .
35
( ) Temperature profile ( ) against
( ) Temperature profile ( ) against
36
( ) Temperature profile ( ) against
( ) Temperature profile ( ) against
( ) Temperature profile ( ) against
37
( ) Temperature profile ( ) against
( ) Temperature profile ( ) against
38
( ) Concentration profile ( ) against
( ) Concentration profile ( ) against
39
( ) Concentration profile ( ) against
( ) Concentration profile ( ) against
40
( ) Concentration profile ( ) against
( ) Concentration profile ( ) against
41
( ) Stream line patterns for
( ) Stream line patterns for
42
( ) Stream line patterns for
( ) ( )
[7] , - , -
: Comparison with the existing literature for the limiting case
43
( ) ( ) ( ) ( )
: Numerical values of non-dimensional quantities when
( ) ( ) ( ) ( )
: Numerical values of non-dimensional quantities when
44
( ) ( )
: Numerical values of non-dimensional quantities when
2.6 Conclusions
Oblique flow of a Casson nano fluid is examined on a convective stretched surface. An efficient
analytical technique OHAM ( ) is utilized to calculate the residual errors of the
governing system of equations. Influence of the flow governing parameters is examined through
45
graphs and Tables. Following conclusions can be drawn from this study:
Heat transfer rate is increasing function of stretching parameter, Prandtl number, Biot
number and it decreases with an increase in Brownian motion, non-Newtonian (Casson)
parameter and thermophoresis.
Nanoparticle concentration is increasing function of stretching parameter and Brownian
motion while it is decreasing function of thermophoresis, Biot number and non-
Newtonian (Casson) parameter.
Nano fluids can be used as an efficient agent to control the heat transfer rate at the
stretching surface.
46
Chapter 3
Combined effects of magnetic field and partial slip on obliquely
striking rheological fluid over a stretching surface
3.1 Introduction
This chapter explores the combined effects of transversely applied uniform magnetic field and
partial slip on an oblique flow of a rheological fluid over stretched convective surface. The
prevailing momentum equations are designed by manipulating Casson fluid model. By applying
the suitable similarity transformations, the governing system of equations are transformed into
coupled nonlinear ordinary differential equations. The resulting system is handled numerically
through midpoint integration scheme together with Richardson’s extrapolation. Streamlines
pattern are presented to study the actual impact of slip mechanism and magnetic field on the
oblique flow. A suitable comparison with the previous literature is provided to confirm the
accuracy of present study for the limiting case.
47
3.2 Mathematical Formulation
Consider a 2D steady flow of a MHD Casson fluid which meets a stretched convective surface in
a non-aligned manner. Surface is stretched along the while taken normal to
surface. A uniformly applied magnetic field is imposed along , which is normal to
the surface as given in ( ) Due to small magnetic Reynolds number the effects of induced
magnetic field are negligible and no electric field is considered. The constitutive equations for
the Casson fluid model are same as in Chapter 2.
( ) Non-dimensional description of the problem
The governing flow equations for the MHD Casson fluid model can be stated as
( )
.
/
( )
48
(
) ( )
where and are components of velocity along and respectively, is the effective
kinematic viscosity and √
is the Casson fluid parameter. The appropriate conditions at
the surface with slip are , -
(
)(
)
( )
in which and are all positive constants with dimensions of inverse time and is a slip
constant.
By applying the similarity transformations
√
√
√
√
√ ( )
Making use of Eq ( ) in Eqs ( ) – ( ) we have
( )
.
/ ( )
(
) ( )
where √
is the Hartman number.
Introducing stream function in the form
( )
49
Eq ( ) is identically satisfied while ( ) ( ) can be written in terms of stream function
as
.
/
( )
( )
( )
Redefining the stream function in Eq ( ) as , -
( ) ( ) ( ) ( )
We get
.
/ ( )
.
/ ( )
Along with transformed boundary conditions
( ) ( ) (
) ( ) ( )
( ) ( ) .
/ ( ) ( )
( )
where and are integral constants to be determined and √ the slip parameter,
Since ( ) (
) at in which signifies the boundary layer displacement constant,
by substituting Eq ( ) in Eqs. ( ) and ( ) we get (
) (
) and
Introducing
( ) ( ) ( )
Eqs ( ) to ( ) take the form as
.
/ (.
/ ) (
) ( )
.
/ ( )
50
( ) ( ) (
) ( )
( )
( ) .
/ ( ) ( ) ( )
Shear stress at the stretching wall in non-dimensional form can be given as
.
/ ( ( )
( )) ( )
The point of stagnation can be attained by zero wall shear stress, i.e.
( )
( ) ( )
where control the obliqueness of the flow.
3.3 Numerical Solution
The governing system of Eqs ( ) to ( ) are coupled and nonlinear in nature which are
solved numerically using highly efficient computational software Maple. Maple uses a
combination of midpoint integration as a basic scheme and Richardson’s extrapolation as an
enhancement scheme, as introduced in , -. This procedure involves a highly stable
computational process which cultivates vastly precise grid independent solutions. The scheme
works by initially transforming the governing nonlinear higher order equations to a set of first
order differential equations. The semi-infinite domain , ) is then replaced with a proper finite
domain of i.e. , ) where is so properly chosen that the numerical solutions would
approach the asymptotic behaviour at infinity. A mesh size of has been supposed,
which is good enough to achieve convergence condition of in almost all computations. Our
51
computed numerical results are compared with the existing published literature in order to certify
the precision of current numerical scheme and are found in decent agreement.
3.4 Results and Discussion
This section helps us to explore the flow behaviour against certain non-dimensional parameters
of interest. ( ) ( ) are deliberated to express the velocities and streamline patterns
for several values of the presiding parameters. ( ) ( ) are plotted to find out the
influence of stretching ratio
Casson fluid parameter slip parameter and magnetic field
parameter on normal velocity component ( ) It is noticeable from ( ) that normal
velocity profile ( ) rises with increasing stretching ratio
and it depicts an inverted boundary
layer behaviour when
From ( ) it is manifested that as Casson fluid parameter is
increasing; normal velocity ( ) drops away from the wall. The behaviour of ( ) against slip
parameter is revealed through ( ) It is evident from this figure that normal
velocity ( ) is inversely related with slip parameter , i.e. it causes the velocity profile
( ) to decrease near the wall. ( ) illustrates the influence of on normal velocity
profile ( ). It is found that in the presence of partial slip, magnetic field parameter causes a
reduction in the normal velocity ( ). This is because of the fact that application of magnetic
field produces lorentz force which causes a resistance within the fluid and hence decreases the
normal fluid velocity. On the other hand, it is quite obvious from ( ) and ( ) that
tangential velocity profile ( ) rises with stretching ratio
and Casson fluid parameter for
fixed values of partial slip parameter and magnetic field parameter . It would be obligatory
52
to mention here that this growth in the tangential velocity profile ( ) is visible near the
stretching wall. Tangential velocity profile ( ) is seemed to be decreasing with an increase in
slip parameter and magnetic field parameter (see ( ) and ( )). In order to observe
the actual flow behaviour in the presence of partial slip and transversely applied magnetic field,
streamline patterns are sketched through ( ) ( ) for positive as well as negative
shear in the flow. It is quite apparent from these graphics that the slip parameter and magnetic
field parameter cause a disturbance in the obliqueness of the flow. However, the effects are
more conspicuous in the presence of magnetic field (see ( ) and ( )). Moreover it is
also noticed that the location of point of stagnation is shifted for slip flow with transversely
applied magnetic field. Numerical values of local skin friction coefficients are presented in
and . offers a suitable agreement of current study with the existing
literature for the limiting case, i.e. in the absence of magnetic field and partial slip when
Numerical values of normal and tangential skin friction components against certain non-
dimensional parameters are presented in . It is found that stretching ratio leads to a
decline in both types of skin friction coefficients. Increasing the magnetic field parameter has
opposite influence on normal and tangential skin frictions. On a similar note it is also found that
slip parameter reduces the skin friction coefficients and increase in Casson fluid parameter
causes an increase in both types of skin friction coefficients.
53
( ) Velocity profile ( ) against .
( ) Velocity profile ( ) against .
54
( ) Velocity profile ( ) against .
( ) Velocity profile ( ) against .
55
( ) Velocity profile ( ) against .
( ) Velocity profile ( ) against .
56
( ) Velocity profile ( ) against .
( ) Velocity profile ( ) against .
57
( ) Streamlines pattern for slip parameter when
( ) Streamlines pattern for slip parameter when
58
( ) Streamlines pattern for magnetic field parameter when
( ) Streamlines pattern for magnetic field parameter when
59
( ) ( )
, - , -
: Comparison with the existing literature when and
60
: Numerical values of local skin frictions against non-dimensional parameters.
3.5 Concluding Remarks
A speculative study is directed to confer the combined effects of partial slip and magnetic field
on an obliquely striking Casson fluid over a stretching surface. The governing system is solved
numerically through midpoint integration scheme with Richardson’s extrapolation. Numerical
values of normal and tangential components of wall shear stress are tabulated.
a/c ( ) ( )
61
These are some of the key findings
Both normal and tangential velocities drops by increasing slip parameter and magnetic
parameter .
Application of magnetic field through parameter has opposite influence on normal and
tangential skin friction components. On the other hand partial slip parameter decays the
skin friction coefficients.
It is exposed from streamline patterns that the point of stagnation is shifted with
partial slip and magnetic field.
62
Chapter 4
Oblique stagnation flow of Jeffery fluid over a stretching
convective surface: Optimal Solution
4.1 Introduction
This chapter is related to the steady flow of Jeffery fluid over a stretching surface with
convective boundary conditions. It is supposed that fluid strikes the wall in an oblique manner.
The flow governing partial differential equations of the non-dimensional problem are presented
and converted to ordinary differential equations by means of suitable similarity transformations.
Optimal homotopy analysis method (OHAM) is operated to deal the resulting ordinary
differential equations. OHAM is found to be extremely effective analytical technique to obtain
convergent series solutions of highly non-linear differential equations. Graphically, non-
dimensional velocities and temperature profile are expressed. Skin friction coefficients and local
heat flux are computed numerically. The comparison of results from this article with the previous
existing literature validates the results of (OHAM) for the limited case.
63
4.2 Mathematical formulation
We considered the steady 2D oblique stagnation point flow of a Jeffery fluid over a stretching
surface as shown in Fig ( ). Further it is assumed that the surface has convective fluid
temperature and uniform ambient temperature (here ( ))
Fig ( ): Schematic diagram of the flow problem
The expression of extra stress tensor for Jeffery fluid is
, (
) - ( )
Where is the Cauchy stress tensor, is the dynamic viscosity, and are the material
constants of the Jeffery fluid and is the Rivlin-Ericksen tensor defined by ( )
( ) .
64
The flow governing equations for two dimensional flow in component form are defined as
( )
(
),
.
/
2 .
/ .
31 ( )
(
),
.
/
2 .
/ .
31 ( )
( )
with the following conditions , -
( )
( )
Where , are positive constant having inverse time dimensions.
In above expressions , are the components of velocity in the and directions,
respectively, is the fluid density, is the kinematic viscosity, is the temperature, the
thermal diffusivity, is the thermal conductivity and is termed as convective heat transfer
coefficient.
Introducing the following quantities
√
√
√
√
( )
Making use of Eq ( ) in Eqs ( ) – ( ), we get
65
( )
(
),
.
/
2 .
/
.
31 ( )
(
),
.
/
2 .
/
.
31 ( )
.
/ ( )
Defining stream function relations as
( )
Substituting Eq ( ) in Eqs ( )– ( ) we get
( ) ( )
( ) ,2.
/ .
/ .
/3
.
/- ( )
.
/ ( )
Along with the boundary conditions
√
( )
( )
Where
represent shear in the stream. We seek solutions of Eqs ( ) and ( ) as
( ) ( ) ( ) ( ) ( )
Replacing Eq ( ) into Eqs ( ) – ( ), after simplification we obtain the following
ordinary differential equations along with the corresponding boundary conditions
66
( )( ) ( ) ( )
( )( ) ( ) ( )
( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( ( )) ( ) ( )
Here Prandtl number non-Newtonian Jeffery parameter and Biot number are defined as
√
( )
Assuming that ( ) .
/ as and making use of Eq ( ) in Eq ( ) we get
( )(
) (Here denotes boundary layer displacement constant). Taking the limit as
in Eq ( ) and by using ( ) we get ( ) Thus Eqs ( )
and ( ) become
( )( (
) ) ( ) ( )
( )( ) ( ) ( )
( )
Note that in above equations when we obtain the equations for viscous fluid.
Introducing
( ) ( ) ( )
Making use of Eq ( ) in ( ), we have
( )( ) ( ) ( )
With the corresponding boundary conditions
( ) ( ) ( )
67
Shear stress and local heat flux at the wall are given in terms of stream function as
.
/ * .
/ .
/+ (
) ( )
In dimensionless form above equations take the following form
.
/ * ( ) ( )
( )+ ( ) ( )
The position of attachment of the dividing stream line is determined by the point of zero wall
shear stress i.e. and can be calculated as
( )
( ) ( ) ( )
4.3 Solution by Optimal Homotopy analysis method
We now solve the nonlinear boundary value problem analytically by OHAM. First of all, it is
quite apparent that ( ) ( ) and ( ) can be written in terms of base functions , -. The
interested reader can refer to the much cited book , - for a systematic and clear exposition on
the HAM.
* ( ) | + ( )
The initial guesses for the governing equations are given by
( )
.
/ ( ( )) ( )
( ) ( ( )) ( )
( ) (
) ( ) ( )
and the auxiliary linear operators are
68
( )
( )
( )
4.4 Optimal convergence-control parameters
Conventional HAM involves a single convergence-control parameter say and it has a
drawback that curves against the convergence controlling parameters (i.e. - curves) cannot
leads towards a particular value which gives the fastest convergent series. Recently, to
fulfill this very deficiency, Liao , - projected the idea of optimal HAM with more than one
convergence-control parameters. He introduced the so-called averaged residual error to get the
optimal convergence parameters efficiently. In general, the optimal HAM can improve
convergence of homotopic solutions. In our problem we are dealing with three coupled non-
linear differential equations, so using (OHAM) means our series solutions contain three non-zero
auxiliary parameters
and to obtain convergence region. In order to find out the total
error as well as the individual error we have optimized the so called local convergence control
parameters
and To find out the optimal values of
and we have used the idea
of average residual error as defined by Liao , -.
∑ *
(∑ ( )
) + ( )
∑ *
(∑ ( ) ∑ ( )
) +
( )
69
∑ *
(∑ ( ) ∑ ( )
) +
( )
Following Liao , -
( )
Where is the total squared residual error, We have considered two cases
when and while other parameters are kept fixed as
Total
average squared residual error is minimized by using computational highly efficient software
Mathematica. We have utilized the idea of to obtain corresponding local optimal
convergence parameters. Tables and are developed to choose the optimal values of so
called optimal convergence control parameters. depicts the minimum value of total
averaged squared residual error, while shows the individual average squared residual
error at different orders of approximations. It can be clearly seen that the averaged squared
residual errors and total averaged squared residual errors goes on decreasing as we increase the
order of approximation. Therefore, one can choose better convergence control parameters
through Optimal Homotopy analysis method to obtain convergent results.
Total averaged squared residual errors for optimal convergence control parameters.
70
Individual squared residual errors for optimal convergence control parameters.
4.5 Results and discussion
The objective here is to analyze effects of emerging non-dimensional parameters of Jeffery fluid
model when the boundary conditions are assumed to be convective. ( ) ( ) are
plotted for the velocity field, temperature field and the stream lines to have a better
understanding of the flow behavior. ( ) and ( ) are plotted to discover the influence of
stretching ratio on velocity components ( ) and ( ) ( ) indicates that increasing
the stretching ratio results in an increase in normal component of velocity ( ).
( ) describes that when
increased, tangential component of velocity ( )is also
increased. ( ) and ( ) depict the variation of another vital
parameter on ( ) and ( ) It is found that with the increase of normal component of the
velocity ( ) decreases, whereas an increase in the tangential component ( ) has been
noticed. ( ) and ( ) are developed to study the influence of non-Newtonian Jeffery
parameter on ( ) and ( ) The behavior was quite contrary to as it was observed for the
71
case of The variation of arising non-dimensional parameters such as stretching parameter
non-Newtonian Jeffery parameter ratio of relaxation to retardation time Prandtl number
and Biot number on temperature profile ( ) are expressed through ( ) – ( ). It is
found that temperature profile ( ) tends to increase with an increase in and Biot number ,
while it depicts a decreasing behavior with an increase in stretching parameter
non-Newtonian
Jeffery parameter and Prandtl number Since the Biot number involves the heat transfer
coefficient, so a higher Biot number means larger heat transfer coefficient and which ultimately
increases the fluid temperature. The stream line patterns for the oblique flows are shown
in ( ) – ( ). The stream line strikes the wall at where is the
stagnation point corresponding to zero wall shear stress. It is noticed from these figures that the
stream line patterns are towards the left of the origin for positive values of whereas it is
on the right of the origin for negative values of
– are presented for local skin friction coefficients and heat flux against emerging
non-dimensional parameters. To confirm precision of our calculated results is
prepared to compare our obtained results with the previous literature for the limited case. From
we have noticed that the magnitude of ( ) decreases with stretching parameter
and where as it exhibits an opposite behavior for the case of ( ) A contrary behavior is
observed for both skin frictions ( ) and ( ) with an increase in Moreover it is also
observed that the shifting of the point of stagnation is dependent upon stretching ratio
retardation time parameter and non-Newtonian Jeffery parameter depicts that
heat transfer ( ) is decreasing function of ratio of relaxation to retardation time parameter
whereas it increases with an increase in Biot number stretching ratio
and non-Newtonian
72
Jeffery parameter It is also noticed that local heat flux at the stretching convective surface
increases for small as well as large values of Prandtl number .
( ): Velocity profile ( ) against .
73
( ): Velocity profile ( ) against .
( ): Velocity profile ( ) against .
74
( ): Velocity profile ( ) against .
( ): Velocity profile ( ) against .
( ): Velocity profile ( ) against .
75
( ): Temperature profile ( ) against .
76
( ): Temperature profile ( ) against .
( ): Temperature profile ( ) against .
77
( ): Temperature profile ( ) against .
( ): Temperature profile ( ) against .
78
( ): Streamlines flow pattern for .
( ): Streamlines flow pattern for .
79
( ): Streamlines flow pattern for .
( ) ( )
[7] , - , -
: Comparison Table against stretching ratio when .
( ) ( )
80
: Numerical values of ( ) ( ) and point of stagnation when .
( )( ) ( )( )
: Numerical values of local heat flux ( )
4.6 Conclusions
We have successfully applied Optimal Homotopy analysis method (OHAM) to achieve series
solutions of the considered problem. We have optimized the so called local convergence control
parameters. Influence of pertinent parameters on velocities and temperature profile have been
depicted graphically. Streamline patterns for several shears in the flow field are expressed
graphically. Local skin friction coefficients, heat flux at the stretching convective surface and
81
numerical values of point of stagnation are calculated and expressed in tabulated form for
small and large Prandtl number .
Following conclusions are drawn from this study:
Non-Newtonian Jeffery fluid parameter has opposite influence on normal and tangential
velocity components whereas it decreases the temperature profile.
Temperature of the fluid decreases with Prandtl number and it enhances with an increase in
Biot number.
Heat transfer increases with an increase in stretching ratio , non-Newtonian Jeffery
parameter , Prandtl number and Biot number .
Comparison with the previous existing literature has been made and considerable agreements
have been found in the results for the limiting case.
82
Chapter 5
A comparative study on non-aligned Jeffery nanofluid over a
stretched surface with heat transfer
5.1 Introduction
This chapter addresses the problem of oblique stagnation point flow using Jeffery nano fluid as a
rheological fluid model. The effects of thermophoresis and Brownian motion are taken into
account. The governing nonlinear partial differential equations of the considered problem are
obtained and then transformed to ordinary differential equations by using suitable
transformations. Consequential highly non-linear system of differential equations are solved
numerically through mid-point integration as a basic scheme along with Richardson’s
extrapolation as an enhancement scheme and analytical results are also obtained using optimal
homotopy analysis Method (OHAM). Non-dimensional velocities, temperature and
concentration profiles are expressed graphically. Local skin friction coefficients, local heat and
mass flux are tabulated numerically as well as analytically for various non-dimensional
parameters emerging in our flow problem. Comparison of the numerical data is made with the
previous existing literature to confirm the accuracy of present study for the case of Newtonian
fluid.
83
5.2 Mathematical Analysis
We have considered steady oblique flow of a Jeffery nano fluid on a stretched surface. Effects of
Brownian motion, thermophoresis and Schmidt number are investigated. All the assumptions are
assumed to be same as considered in previous . The constitutive equations of mass and
motion for Jeffery fluid model are given in ( )– ( ) while the energy and nanoparticle
concentration equations are given in Eqs ( ) ( ). The resulting equations of flow,
temperature and concentration of nanoparticles are given by
( )
(
),
.
/
2 .
/ .
31 ( )
(
),
.
/
2 .
/ .
31 ( )
( )
( ) (
) ( )
( )
The corresponding suitable conditions are , -
( )
84
In above expressions and are the components of velocity along and perpendicular to the
surface respectively, are the fluid and nanoparticle mass density, is kinematic viscosity,
is the temperature, is the concentration, is the specific heat of the material and is the
thermal diffusivity of the fluid.
Introducing the following quantities
√
√
√
√
( )
Eqs ( ) ( ) become
( )
(
),
.
/
2 .
/
.
31 ( )
(
),
.
/
2 .
/
.
31 ( )
(
) ( ) ( )
.
/
( )
By applying the stream functions relations from Eq ( ) and similarity solutions from
Eq ( ) in above, we get,
( )( ) ( ) ( )
( )( ) ( ) ( )
85
(
) ( )
( )
( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ( )
where are integral constants to be determined. Here Prandtl number the non-
Newtonian Jeffery parameter the thermophoresis parameter the Brownian motion
parameter and the Schmidt number are defined as
( )
( )
( )
( )
( )
( )
( )
Assuming that ( ) .
/ as and using Eq ( ) we get ( )(
) and
( ) thus we get
( )( (
) ) ( ) ( )
( )( ) ( ) ( )
(
) ( )
( )
Introducing
( ) ( ) ( )
Using Eq ( ) in ( ), we have
( )( ) ( ) ( )
With the boundary conditions
( ) ( ) ( )
86
5.3 Non-dimensional quantities of Interest
Shear stress and local heat and mass flux at the wall are given in terms of stream function as
(
) { (
)4
5}
(
) (
) ( )
In transformed form above equations take the following form
.
/ * ( ) ( )
( )+ ( ) ( ) ( )
The position of attachment of the dividing stream line is determined by the point of zero wall
shear stress i.e. and can be calculated as
( )
( ) ( ) ( )
5.4 Method of Solution
5.4.1 Numerical solution
By means of similarity transformations, the governing equations ( ) ( ) are converted to
nonlinear differential equations ( ) ( ) along with their appropriate boundary
conditions. This system is then solved numerically using mid-point integration approach to avoid
certain singularities appearing in the flow governing equations (see for example , -). The
governing system is solved on a sufficiently large but finite domain of with a mesh size
of until we achieved the desired degree of accuracy, called .
87
5.4.2 Analytical solution by Optimal Ham
In order to verify our numerical results, the governing equations ( ) – ( ) are also solved
analytically through optimal HAM. The detailed procedure of OHAM has already been provided
in . Here we will provide the total and individual averaged squared residual errors.
The initial guesses and of ( ) ( ) ( ) and ( ) are selected as follow
( )
.
/ ( ( )) ( )
( ) ( ( )) ( )
( ) ( ) ( )
( ) ( ) ( )
We select the auxiliary linear operators as
( )
( )
( )
( )
5.4.3 Optimal convergence-control parameters
To determine suitable values of convergence parameters
and we have utilized the
idea of so-called Average Square Residual Error defined by , -.
∑ *
(∑ ( )
) + ( )
88
∑ *
(∑ ( ) ∑ ( )
) +
( )
∑ *
(∑ ( ) ∑ ( )
) +
( )
∑ *
(∑ ( ) ∑ ( )
) +
( )
Following Liao , -
( )
Where is the total squared residual error, we have considered a case
where
Total average squared residual
error is minimized by using Mathematica package . and are prepared
for the case of several optimal convergence control parameters. shows the minimum
value of total averaged squared residual error, while shows the individual average
squared residual error at different orders of approximations using the optimal values from
at . We can observe that the averaged squared residual errors and total
averaged squared residual errors goes on decreasing with higher order approximations. Besides
these Tables, graphical results for velocities, temperature, concentration and averaged squared
residual error are also provided through ( ) and ( ). From these figures we can see that
our obtained results are very much correct and it is also noticed that the total averaged squared
residual error decreases as we increase the number of iterations (see ).
89
, -
: Total Averaged squared residual errors using
CPU TIME[S]
: Individual Averaged squared residual errors using optimal values at from
.
90
( ) Graphical comparison for velocities, temperature and concentration profiles using
Numerical scheme (Right) and OHAM (Left).
( ) Total averaged squared Residual Error against no of iterations .
91
5.5 Results and discussion
This section deals with the graphical description of velocity, temperature and concentration
profiles for several emerging non-dimensional parameters in our flow problem.
( ) – ( ) are plotted for this purpose. ( ) ( ) are plotted to discover
the influence of stretching ratio
and non-Newtonian Jeffery parameter on normal and
tangential velocity components i.e. ( ) and ( ) ( ) indicates that increasing the
stretching ratio
and non-Newtonian Jeffery parameter results in an increase in the normal
component of the velocity ( ). ( ) describes that when
increases; tangential
component of velocity ( ) also increases. We can also notice that when
increasing the
non-Newtonian Jeffery parameter decreases the tangential component of
velocity ( ) until ( ) but after that the behavior is reversed. On the other hand this
behavior is found to be quite contrary when stretching ratio
( ) and ( )
exhibit the influence of non-dimensional parameter which is the ratio of relaxation to
retardation time on ( ) and ( ). It is found that with increasing ratio normal component
of the velocity ( ) decreases where as an increase in the tangential component ( ) has been
noticed when
but this tendency is reversed when
Influence of non-dimensional
parameters such as thermophoresis parameter Brownian motion parameter Prandtl
number and Schmidt number on the temperature profile ( ) and concentration profile
( ) are discussed through ( ) – ( ). It is depicted from ( ) and ( ) that
temperature profile ( ) and concentration profile ( ) tends to increase with an increase in
thermophoresis parameter The corresponding thermal and diffusion boundary layer thickness
92
also increase with an increase in thermophoresis parameter It is noticed from ( ) and
( ) that Brownian motion parameter has opposite influence on the temperature profile
( ) and concentration profile ( ) Increasing the Brownian motion parameter enhances
the temperature profile ( ) and the corresponding thermal boundary layer thickness
appreciably increases but on the other hand concentration profile ( ) as well as the related
diffusive boundary layer thickness decays significantly with an increase in Brownian motion
parameter This is due to the fact that an increase in the Brownian motion leads to an increase
in more random collisions with in the fluid particles which consequently heats up the fluid and
raises its temperature and decreases its concentration. ( ) describes that temperature
profile ( ) drops prominently and thermal boundary layer thickness rapidly decreases for large
values of Prandtl number , while it is quite evident from ( ) that when Prandtl
number is increased; concentration profile ( ) initially increases and then goes on
decreasing away from the surface. Finally the influence of Schmidt number on the
temperature profile ( ) and concentration profile ( ) are presented through
( ) and ( ). The behaviour of both these non-dimensional quantities against
Schmidt number is found to be opposite, i.e. it enhances the temperature profile ( ) while
decrease the concentration profile ( ). Interestingly, it is observed from ( ) that for
large values of Schmidt number ; diffusion boundary layer thickness decreases abruptly.
– are presented for the numerical values of components of shear stress at the
wall, local heat flux and mass against emerging parameters of interest. provides a
comparison with the existing results for the limited case and it shows that our computed results
are in very good agreement. From it is noticed that absolute values of the normal
component of shear stress ( ) declines with an increase in stretching parameter
and Jeffery
93
parameter and increases with an increase in the ratio of relaxation to retardation time
parameter where as it exhibits an opposite behavior for the case of stretching parameter
and
Jeffery parameter for the tangential skin friction coefficient ( ) We have observed that
has no substantial effect on skin friction coefficient component ( ) One can also observe very
easily that local heat flux ( ) and mass flux ( ) are reduced with an increase of
stretching ratio
Local heat and mass flux are responding in an opposite manner for both non-
Newtonian Jeffery parameter and ratio of relaxation to retardation time shows
that local heat flux ( ) is decreasing function of thermophoresis parameter Brownian
motion parameter and Schmidt number while it decreases with Prandtl number Local
mass flux ( ) at the wall increases with an increase in Brownian motion
parameter Prandtl number Schmidt number and thermophoresis parameter
( ): Velocity profile ( )against
94
( ): Velocity profile ( )against
95
( ): Velocity profile ( )against
( ): Velocity profile ( )against
96
( ): Temperature profile ( ) against
( ): Concentration profile ( ) against
97
( ): Temperature profile ( ) against
( ): Concentration profile ( ) against
( ): Temperature profile ( ) against
98
( ): Concentration profile ( ) against
( ): Temperature profile ( ) against
99
( ): Concentration profile ( ) against
( ) ( ) ( )
[7] , - , - , -
Comparison Table for when
100
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
: Numerical values of non-dimensional constant ( ) , ( ) and ( )
, ( ) when
( ) ( ) ( ) ( )
101
: Numerical values of local heat flux ( ) and local mass flux ( ) when
5.6 Conclusions
This numerical study is conducted to highlight the problem of oblique flow of a rheological
Jeffery fluid model under the influence of nanoparticles. Normal and tangential components of
local shear stress, local heat and mass flux are calculated and expressed in tabulated form.
These are few of the key findings.
Local skin friction coefficients exhibits a contrary behavior with an increase in stretching
ratio while local heat flux as well as local mass flux reduces by an increase in stretching
ratio
It is found that heat flux ( ) is increasing function of non-Newtonian Jeffery parameter
and Prandtl number
Thermophoresis and Brownian motion decays heat flux ( ) at the wall.
The effects of Brownian motion and thermophoresis on the local mass flux ( )
are opposing each other.
Comparison with the previous existing literature has been made and considerable agreement
has been found in the results for the limiting case.
102
Chapter 6
MHD second grade oblique flow over a convective surface with
soret and dufour effects
6.1 Introduction
This chapter addresses the soret dufour effects on a steady, 2-D magneto hydrodynamic (MHD)
flow of an obliquely striking second grade fluid over a convective surface. The governing
systems of equations are extremely complicated and non-linear in nature which are tackled by
midpoint integration scheme with Richardson’s extrapolation. The results for normal and
tangential velocities, temperature and concentration are analysed. Numerical computations are
performed to discover the influence of pertinent non-dimensional parameters on normal and
tangential components of local shear stress, local heat flux and local mass flux. It is found that
magnetic field disturbs obliqueness of the flow, decreases the normal skin friction co-efficient
whereas it enhances the tangential skin friction co-efficient. Influences of soret number and
dufour number on temperature and concentration are found to be quite opposite. A suitable
agreement with the existing literature is shown to endorse the current study for the limiting case.
6.2 Mathematical Formulation
Consider a steady, 2D MHD oblique viscoelastic fluid which meets stretched convective surface
in a non-aligned manner. Surface is made to keep stretched along the . Magnetic field of
uniform strength is imposed along , which is normal to the surface which is
presented through ( ).
103
The Cauchy stress tensor for second grade viscoelastic fluid is given by
( )
Where and are first and second Rivlin-Ericksen tensors given by ( ) ( )
and
( ) ( ) .
( ): Non-dimensional description of the problem
The equations which govern the flow problem are , -
( )
{
[
.
/
.
/]
0.
/ .
/
1}
[ .
/
.
/
]
( )
{
0.
/ .
/
1
[
.
/
.
/]}
[ .
/
104
.
/
] ( )
( )
( )
where are and components of the velocity respectively, is the temperature, is the
concentration, is the mass diffusivity, is the effective kinematic viscosity, is the thermal
diffusion, is the specific heat, is the concentration Susceptibility, is the constant applied
magnetic field, is the fluid mean temperature and is the effective thermal diffusivity of the
MHD viscoelastic fluid.
The boundary conditions of the problem are , -
( )
( )
in which are all constants having inverse time dimension, is the convective heat
transfer coefficient, is the constant concentration at the wall whereas , are the ambient
temperature and concentration respectively.
By applying the similarity transformations , -
√
√
√
√
√
( )
Eqs ( ) ( ) give
( )
{
[
.
/
.
/]
105
0.
/ .
/
1}
[ .
/
.
/
] ( )
{
0.
/ .
/
1
[
.
/
.
/]}
[ .
/
.
/
] ( )
( )
( )
where
is the Weissenberg number, √
is the Hartman number,
is
the Prandtl number,
is the Soret number,
is the Dufour
number,
is the Schmidt number,
√
is the Biot number and
.
Introducing the stream function relations
( )
Using Eq ( ), Eqs ( ) ( ) can be written in terms of stream function as
( )
( )
( )
( )
( )
( )
( )
We search for solution of Eqs ( ) ( ) of the form
( ) ( ) ( ) ( ) ( ) ( )
This gives
( ) ( )
106
( ) ( )
( ) ( )
( ) ( )
Along with the transformed boundary conditions
( ) ( ) ( )
( ) ( ) ( )
( ) ( ( )) ( ) ( ) ( ) ( )
Since ( )
at in which signifies the boundary layer displacement constant.
Substituting Eq ( ) in above, we get (
) (
) and
Introducing
( ) ( ) ( )
Equations ( ) ( ) take the form
( ) ((
) ) (
) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( ( )) ( ) ( ) ( ) ( )
The local shear stress, local heat flux and local mass flux at the stretching convective surface, are
given as , -
,
*.
/
.
/
+-
.
/
.
/
( )
107
Employing Eq ( ), above quantities finally take the form as,
*( )+ ( ) *( )+ ( )
( ) ( ) ( )
The point of stagnation can be attained by zero wall shear stress, i.e.
*( )+
( )
*( )+ ( ) ( )
6.3 Numerical Solution
The systems of equations ( ) – ( ) are solved numerically utilizing midpoint integration
scheme coupled with Richardson’s extrapolation. Midpoint technique is an extremely stable
computational process which when merged with Richardson’s extrapolation develops highly
accurate grid independent solutions. The problem of semi-infinite domain , ) is generously
transformed into such appropriate finite domain of i.e. , ) where should be so large that
the numerical solutions would approach the asymptotic behaviour at the infinite boundaries. A
mesh size of has been selected to achieve convergence up to in our
computations. Numerical computations are performed on a symbolic highly efficient software
Maple.
6.4 Results and Discussion
This segment illuminates the flow behaviour against pertinent non-dimensional parameters
appearing in the governing system. Figs. ( )– ( ) are plotted to express the velocity,
temperature and concentration profile of the system. Stream line patterns are sketched to find out
108
the impact of applied magnetic force through parameter on oblique flow. Figs ( ) – ( )
are portrayed to find out the effect of magnetic field , stretching ratio and Weissenberg
number on velocity profiles ( ) and ( ). It is found that the Hartman number causes
to decrease both types of velocity profiles, which is obvious from Figs ( ) and ( ). We can
notice from Fig ( ) that when , increase in magnetic field causes to decrease the
normal component of the velocity ( ). This is because Lorentz force which is developed
through the application of uniform magnetic strength executes a retarding force on velocity
profiles and hence causes a reduction in the velocities. On the other hand, it is quite evident that
stretching ratio and viscoelastic parameter has positive impact on the velocities. From
Fig ( ), we observe that when the magnetic field parameter is kept fixed and as the
viscoelastic parameter increases, normal component of velocity ( ) also increases.
Similar kind of behaviour is noticed in the case of tangential component of velocity ( ) (see
Fig ( )). The behaviour of temperature profile ( ) against certain non-dimensional
parameters is expressed through Figs. ( ) – ( ). Fig ( ) exhibit that with an increase in
magnetic field temperature profile ( ) rises extensively. This behaviour is observed because
of the resistive force which consequently heats up the fluid and rises its temperature. On the
other hand, it is observed from Fig ( ) that increasing the viscoelastic parameter results in
an increase in the temperature profile ( ). From Fig ( ), it is noticed that when Biot number
increases, temperature profile ( ) also increases. This is because Biot number involves
the heat transfer coefficient so an increase in Biot number leads to an increase in heat transfer
coefficient which consequently enhances the fluid temperature ( ). Effects of Soret number
and Dufour number on the temperature profile ( ) are depicted through Figs ( )
and ( ). These parameters have shown an opposite effect on temperature profile ( ) for fix
109
values of Hartman number . It is found that with an increase in Soret number , temperature
profile ( ) decreases near the wall (see Fig ( )). Soret number is the ratio of temperature to
the concentration difference so an increase in it leads to a higher temperature difference at the
wall and the ambient fluid temperature compared to the corresponding concentration difference,
due to which temperature drops down. Similarly we can observe from Fig ( ) that Dufour
number causes to increase temperature profile ( ) significantly. Dufour number is the ratio
of concentration to temperature difference. Higher values of Dufour number implies lower
temperature difference, which results in an enhancement in the temperature profile ( ).
Similarly, Fig ( ) reveals that when we increase the value of Schmidt number ,
temperature rises under the influence of uniform magnetic force. The influence of important
parameters like magnetic field parameter , viscoelastic parameter , Biot number Dufour
number Soret number and Schmidt number on ( ) is presented through
Figs ( ) – ( ). One can perceive from Fig ( ) that magnetic field parameter
increases the concentration profile ( ). Increasing the viscoelastic parameter leads to a
decrease in concentration profile ( )which is visible from Fig ( ). Effect of Biot number i
on the concentration profile ( ) is depicted through Fig ( ). We can see that concentration
profile ( ) is increasing with Biot number . It is interesting to note here that Biot number has
similar kind of influence on the concentration profile ( ), which it have on the temperature
profile ( ) but the effects are more prominent in temperature profile ( ) when compared with
the concentration profile ( ). Impact of pertinent parameters such as Soret and Dufour
number on the concentration profile ( ) is presented through Figs ( ) and ( ). Both
these parameters have opposite influence on the concentration profile ( ) near the wall.
Similarly, we can notice from Fig ( ) that for fixed magnetic field, concentration profile
110
( ) drops with Schmidt number . Stream line patterns for oblique flow under applied
magnetic field are traced through Figs ( ) and ( ). It is easily detectable from these
stream lines that the magnetic field parameter caused the obliqueness of the flow to slightly
straighten up. This is because normal shear flow is disturbed through the application of magnetic
force of uniform strength. Comparison of present study with the previously existing literature
when ( ) is given in . The computed results are found to be in decent agreement
with those given by Pop et al , - Variations of local skin frictions co-efficient ( ) and
( ) local heat flux ( ) and local mass flux at the surface ( ) against magnetic field
parameter , stretching ratio and Weissenberg number are presented through .
It is found that normal component of the local skin frictions co-efficient ( ) as well as
tangential component of the local skin friction co-efficient ( ) decreases with an increase in
stretching ratio . An increase in stretching parameter consequently favours the stretch flow
which ultimately reduces the skin friction at the wall. On the other hand it is also noticed that
heat flux at the convective surface as well as the local mass flux increases when stretching
ratio
is increased. This is in good agreement with the non-dimensional situation because
increasing the stretching ratio decreases the corresponding surface temperature due to which
heat diffuses quickly at the surface and the corresponding mass flux increases. Viscoelastic
parameter has an opposite influence on normal skin friction co-efficient ( ) and
tangential skin-frictions ( ) respectively. It is observed that increasing the viscoelasticity
parameter , causes an increase in local heat flux ( ) and mass flux ( ). The effect of
magnetic field on normal and tangential skin friction co-efficient, i.e. ( ) and ( ) is not
very similar. The obtained numerical data indicates that with increasing the magnetic field
strength through parameter , normal component of the skin friction ( ) rises, while
111
tangential skin friction ( ) gradually drops. As we have noticed earlier that magnetic field
raises the fluid temperature and concentration profiles so consequently local heat and mass flux
at the wall decays which is quite obvious from the obtained numerical results in .
is presented to find out the influence of relevant non-dimensional parameters such as
Soret number , Prandtl number , Dufour number Schmidt number and Biot number
on local heat ( ) as well as mass flux at the surface ( ). One can easily observe
from the tabulated results that Biot number Soret number and Prandtl number caused
an increase in the local heat flux ( ), while they lead to a reduction in the local mass
flux ( ). Similarly for large values of Dufour number and Schmidt number , local heat
flux rate ( ) declines whereas local mass flux rate ( ) at the stretching convective
surface improves in a quantitative sense.
Fig ( ): Velocity profile ( ) against .
112
Fig ( ): Velocity profile ( ) against .
Fig ( ): Velocity profile ( ) against .
113
Fig ( ): Velocity profile ( ) against .
Fig( ): Temperature profile ( ) against .
114
Fig( ): Temperature profile ( ) against .
Fig( ): Temperature profile ( ) against .
115
Fig( ): Temperature profile ( ) against .
Fig( ): Temperature profile ( ) against .
116
Fig( ): Temperature profile ( ) against .
Fig ( ): Concentration profile ( ) against .
117
Fig ( ): Concentration profile ( ) against .
Fig ( ): Concentration profile ( ) against .
118
Fig ( ): Concentration profile ( ) against .
Fig ( ): Concentration profile ( ) against .
119
Fig ( ): Concentration profile ( ) against .
Fig ( ): Stream line flow pattern for and
120
Fig ( ): Stream line flow pattern for and
( ) ( ) ( )
, - , - , -
: Comparison with the existing literature when
121
: Numerical values of local skin frictions and heat
and mass flux against
and .
a/c A ( ) ( ) ( ) ( )
122
: Numerical values of local heat and mass flux against and .
( ) ( )
123
6.5 Concluding Remarks
A theoretical study is conducted to discuss the MHD oblique viscoelastic fluid over a stretching
convective surface. The governing system of equations is integrated by midpoint scheme along
with Richardson’s extrapolation. Numerical quantities such as local shear stress, local heat as
well as mass flux are tabulated.
The key points of study can be stated as.
i. Impact of uniform magnetic strength through non-dimensional parameter on normal and
tangential velocity components has been observed similar. I.e. it causes a reduction in both
types of velocities.
ii. Both types of velocities exhibit an increasing behaviour with increasing stretching ratio
and viscoelastic parameter .
iii. Temperature and concentration profiles substantially grow with an enhancement in magnetic
field parameter .
iv. Soret and Dufour numbers has an opposite impact on temperature as well as concentration
profile for fixed magnetic strength.
v. Application of magnetic field through Hartman number leads to a disturbance in the
obliqueness of the flow.
124
Chapter 7
Numerical investigation on MHD oblique flow of Walter’s B
nano fluid over a convective surface
7.1 Introduction
The objective of this chapter is to numerically investigate the oblique flow of a Walter-B nano
fluid over a convective surface. Effects of transversely applied magnetic field are also taken into
account. The governing system is presented in the form of coupled differential equations by
means of suitable similarity transformations which are then solved by using Spectral
Quasilinearization Method (QLM) and the Spectral Local Linearization Method (LLM). The
results for velocities temperature as well as nano particle concentration are plotted against
pertinent flow parameters. It is found that applied magnetic field has opposite influence on
normal and tangential components of local shear stress and it decays the local heat flux and mass
flux rate at the stretching convective surface. Thermophoresis and Brownian diffusion effects on
the local heat and mass flux rate are found to be non-similar in a quantitative sense. In order to
signify the validity of current numerical scheme, a remarkable agreement is presented with the
previous literature for the limiting case.
125
7.2 Mathematical Formulation
Let us consider oblique flow of a Walter-B nanofluid which strikes a convective surface. A
magnetic field of uniform strength is executed in the direction of . Moreover, surface
is kept stretched by applying two equal and opposite forces making the origin fixed (as shown in
Fig ( )). We further assume that the surface has convective fluid temperature and uniform
ambient temperature (here ( )) The constitutive equations for the present problem are
the same as ( ) ( ). The stress tensor for Walter B type fluid can be written as , -
( )
Where the short memory co-efficient, is the deformation rate tensor and
is upper
convected derivative of a tensor and can be obtained by
( )
Using the constitutive equations ( ) ( ) and the extra stress tensor for Walter B type
fluid from equation ( ), the flow governing equations are given as , -
Fig ( ): A non-dimensional description of the problem
126
( )
2.
/
0
(
)
13
( )
2.
/
0
(
)
13 ( )
( )
( ) (
) ( )
( )
where , are the components of velocity along co-ordinate axis, is the effective kinematic
viscosity , is the temperature, is the concentration, is the effective thermal diffusivity,
is the Brownian diffusion co-efficient and is the thermophoresis co-efficient.
The appropriate boundary conditions for the problem under consideration are , -
( )
( )
in which are all constants having inverse time dimension, is the convective heat
transfer coefficient, is the constant concentration at the wall whereas and are ambient
fluid temperature and concentration.
By applying
√
√
√
√
√
( )
127
Eqs ( ) ( ) give
( )
2.
/
0
(
)
13 ( )
2.
/
0
(
)
13 ( )
(
) ( ) ( )
(
)
( )
Where
is the local Weissenberg number, √
is the Hartman number,
is
the Prandtl number, the thermophoresis parameter
( )
( )
( )
the Brownian motion
parameter
( )
( )
( )
and
is the Schmidt number.
Introducing the stream function relations
( )
Eqs ( ) ( ) can be written in terms of stream function as
( )
( )
( )
( )
( )
(
) ( ) ( )
(
)
( )
We search for solution of Eqs ( ) ( ) of the form
128
( ) ( ) ( ) ( ) ( ) ( )
This gives
( ) ( )
( ) ( )
(
) ( )
( )
Along with the transformed boundary conditions
( ) ( ) ( )
( ) ( ) ( )
( ) ( ( )) ( ) ( ) ( ) ( )
Since ( )
at here signifies the boundary layer displacement constant and
√
is the Biot number. Substituting Eq ( ) in above, we get (
) (
)
and
Introducing
( ) ( ) ( )
Equations ( ) ( ) along with the boundary conditions ( ) take the form
( ) (.
/ ) (
) ( )
( ) ( )
(
) ( )
( )
( ) ( ) ( )
( ) ( )
129
( ) ( ( )) ( ) ( ) ( ) ( )
Local Shear stress and heat, mass flux at the stretching convective surface in non-dimensional
form are given by
( ) ( ) ( ) ( )
( ) ( ) ( )
The point of stagnation can be attained by zero wall shear stress, i.e.
( )
( )
( ) ( )
( )
Where control the obliqueness of the flow.
7.3 Numerical Solution
Equations ( ) ( ) with the boundary conditions ( ) constitute a system of nonlinear,
non-homogeneous differential equations for which closed form solution cannot be found. Hence,
these equations are tackled through Spectral Quasilinearization Method (QLM) and the Spectral
Local Linearization Method (LLM). The Quasilinearization technique (QLM) is essentially a
generalized Newton-Raphson Method that was originally used by Bellman and Kalaba , - for
solving functional equations. Considering that the momentum equation ( ) is decoupled from
the rest of the equations in the system, this equation is solved for ( ) first using the QLM.
When the approximate solution for ( ) is substituted in equation ( ), the equation becomes
linear and is solved directly by the Chebyshev spectral collocation scheme. Applying the QLM
on ( ) ( ) we obtain the following iterative sequence of linear equations;
( )
130
( )
( )
( )
Subject to
( ) ( ) ( )
( ) ( ( )) ( )
( )
( ) ( ) ( ) ( ) ( )
Where
( ) (
( ) ) .
/ .
/
(
( ) )
Equations ( ) and ( ) can be solved independently for ( ) and ( ). However, ( )
and ( ) constitute a linear system of coupled differential equations with variable coefficients
and must be solved simultaneously. The entirely system can be solved iteratively using any
numerical method for In this work, the Chebyshev spectral collocation method
approach will be utilized as describe in , -. Starting from a given set of initial
approximations the iteration schemes ( ) ( ) can be solved iteratively
for ( ) ( ) ( ) and ( ) when before applying the spectral
131
collocation method, the domain in is transformed to the interval , - so that spectral method
can be applied. For the convenience of the numerical computations, the semi-infinite domain in
is approximated by the truncated domain , -, where is a finite number selected to be large
enough to represent the behaviour of the flow properties when is very large. We use the
transformation ( ) to map the interval , - to , -. The basic idea behind the
spectral collocation method is the introduction of a differentiation matrix which is used to
approximate unknown derivatives of variables ( ) at the grid points as the matrix vector
product
( )
|
∑ ( ) ( )
Where is the number of collocation points,
, ( ) ( ) ( )-
Is the vector function at the collocation points. Similar vector functions corresponding to and
are denoted by and respectively. Higher order derivatives are obtained as powers of D,
that is
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
Where is the order of the derivative. The matrix have the size ( ) ( ) The grid
points on are defined as
( )
132
Where , represents total grid points in direction. Thus, applying the spectral method on
equations ( ) ( ) give
( )
( )
[
] 0
1 [
] ( )
Where and ( ) are ( ) ( ) matrices defined as:
( )
( )
( )
( )
and ( ) are ( ) vectors obtained by evaluating the right hand sides of
( ) ( ) at the collocation points.
7.4 Graphical Results and Analysis
To find out the impact of certain emerging flow parameters on the normal and tangential
velocities, temperature and concentration, Figs ( ) ( ) are plotted. The influence of
applied magnetic field through parameter on normal component of flow ( ) and velocity
profiles ( ) is shown through Figs ( ) and ( ). It is quite obvious from these two figures
that when , flow and velocity profiles are higher near the wall and it continuously drops
133
down with increasing magnetic field . These is because of upward direction of applied force
which opposes the normal flow and velocity profiles, consequently leads to a significant drop in
normal component of flow and velocity profiles. From Figs ( ) and ( ) we can perceive that
tangential component of the flow ( ) as well as magnitude of the tangential velocity profile
( ) declines with Magnetic field . With this, we can come up to the conclusion that
application of Magnetic field causes the normal as well as tangential flow and velocity profiles to
decrease near the convective surface. The effect of applied magnetic field on temperature
( ) and concentration profiles ( ) is depicted through Figs ( ) and ( ). We can see that
the influence of magnetic field on the temperature profile ( ) is positive. It significantly rises
with increasing magnetic field. While on the other hand, the behaviour of the concentration
profile ( ) against can be seen from Fig ( ). It is observed that concentration profile ( )
increases with . Figs ( ) and ( ) are plotted to discover the influence of thermophoresis
parameter on the temperature ( ) and concentration profile ( ) respectively. It can be
noticed here that with increasing , temperature profile ( ) rises within the boundary layer
region (see Fig ( )). Impact of thermophoresis on the concentration profile ( ) is
different from as it was on temperature. We can see that it slightly drops down very close to the
wall, and then depicts an increasing behaviour with increasing . Similarly the effects of
Brownian motion , on the temperature ( ) and concentration profile ( ) is revealed
through Figs ( ) and ( ). It is quite evident from these graphs that Brownian motion has
opposite influence on temperature ( ) and concentration profile ( ), i.e. it causes the
temperature to rise down while leads to a significant drop in the concentration profile. This is
perhaps due to the reason that the random motion of the particles results in continuous collisions
of the fluid particles which consequently heats up the fluid and drops the concentration profile.
134
Figs ( ) and ( ) are constructed to examine temperature ( ) and concentration profile
( ) versus Prandtl number respectively. We can see that temperature decreases with Prandtl
number (see Fig ( )). On the other hand, concentration profile ( ) gives an increasing
behaviour with within the boundary layer region (See Fig ( )). We can notice from Fig
( ) that temperature profile ( ) rises with an increase in Schmidt number . But when we
look at the graph of concentration profile ( ) through Fig ( ), it is quite apparent that
concentration profile ( ) significantly drops down with . Finally, effects of Biot number ,
on the temperature and concentration profiles are presented through Figs ( ) and ( ). One
can certainly realise that influence of Biot number on these two quantities is positive, i.e. it
causes an increase in temperature ( ) and concentration profile ( ) near the stretching
surface. Finally, Fig ( ) is plotted to discover the stream line patterns when and
It can be seen that application of magnetic field causes a disturbance in the
obliqueness of the flow. gives the numerical values of ( ) obtained by Pop et al
, - and are compared with our numerical results when and they are in very good
agreement. gives the variations in boundary layer displacement constant with
and . Similarly and are prepared to compute the normal ( ) and tangential
( ) components of local shear stress respectively. It is observed from that with an
increase in magnetic parameter ( ) also increases. It is found from that
magnetic field parameter causes to reduce the tangential skin friction component ( ) This is
perhaps because of the upward direction of the constantly applied magnetic field which favours
the shear flow and ultimately reduces the tangential skin friction ( ). shows that
with an increase in the magnetic field through parameter , local heat flux ( ) at the wall
decreases. It can also be noticed from this table that increasing the viscoelasticity through
135
parameter , causes a reduction in local heat flux ( ) at the wall. The influence of
thermophoresis parameter , Prandtl number Brownian motion and Schmidt number
on the local heat flux ( ) is presented through and . It is noticed from
that local heat flux drops with Brownian motion as well as thermophoresis
parameter . Similar kind of behaviour is observed for the local heat flux with increasing
Prandtl number as well as Schmidt number see e.g. ( ). The main reason behind it
is that a larger Prandtl number fluid means lower thermal conductivity which consequently
decreases heat transfer rate at the surface. Influence of viscoelasticity through parameter ,
magnetic field parameter , thermophoresis parameter , Prandtl number Brownian
motion and Schmidt number on the local mass flux ( ) are presented through
We can easily see from that when and increasing the
magnetic field parameter , the local mass flux ( ) decreases. But for all other larger values
of , magnetic field causes the local mass flux ( ) to rise at the stretching convective
surface. Similarly from it can be perceived that for fixed values of Brownian motion
parameter , local mass flux rate ( ) decreases with thermophoresis parameter Finally,
it is found from that local mass flux ( ) also increases with when Prandtl
number is fixed.
136
Fig ( ): Flow profile ( ) against when
Fig ( ): Velocity profile ( ) against when
137
Fig ( ): Flow profile ( ) against when
Fig ( ): Velocity profile ( ) against when
138
Fig ( ): Temperature profile ( ) against when
Fig ( ): Concentration profile ( ) against when
139
Fig ( ): Temperature profile ( ) against when
Fig ( ): Concentration profile ( ) against when
140
Fig ( ): Temperature profile ( ) against when
141
Fig ( ): Concentration profile ( ) against when
Fig ( ): Temperature profile ( ) against when
142
Fig ( ): Concentration profile ( ) against when
Fig ( ): Temperature profile ( ) against when
143
Fig ( ): Concentration profile ( ) against when
Fig ( ): Temperature profile ( ) against against when
144
Fig ( ): Concentration profile ( ) against when
Fig ( ): Stream line patterns for oblique flow when
145
( ) ( ) ( )
, - , - , -
: Comparison with the existing literature when ,
M\Kw 0 0.1 0.2 0.4 0.6 0.8
0 0.32860 0.28959 0.24594 0.14574 0.13698 0.13376
0.1 0.32791 0.28899 0.24543 -0.76143 0.13808 0.13111
0.2 0.32587 0.2872 0.2439 0.21167 0.14156 0.12417
0.4 0.31809 0.28033 0.23806 0.17889 0.15757 0.10837
0.6 0.30622 0.26988 0.22917 0.26588 0.19058 0.10908
0.8 0.29158 0.25698 0.2182 0.12677 0.24908 0.13853
: Numerical values of boundary layer displacement constant when
M\Kw 0 0.1 0.2 0.4 0.6 0.8
146
0.0 -0.66726 -0.73000 -0.81021 -0.45861 -0.56181 -1.00182
0.1 -0.66910 -0.73203 -0.81251 -4.05864 -0.55023 -0.99097
0.2 -0.67459 -0.73810 -0.81937 0.69801 -0.51556 -0.95877
0.4 -0.69614 -0.76193 -0.84628 0.08536 -0.37786 -0.83487
0.6 -0.73071 -0.80014 -0.88942 0.86451 -0.15032 -0.64388
0.8 -0.77665 -0.85088 -0.94664 -1.18104 0.16756 -0.39986
: Numerical values of Normal skin friction co-efficient ( ) when
M\Kw 0 0.1 0.2 0.4 0.6 0.8
0.0 0.78268 0.74961 0.70756 0.60383 0.38110 0.06251
0.1 0.69085 0.66268 0.62736 0.54442 0.33658 0.05022
0.2 0.45858 0.44312 0.42509 0.35539 0.22104 0.01829
0.4 -0.05075 -0.03617 -0.01496 -0.03014 -0.06444 -0.06160
0.6 -0.27686 -0.24785 -0.20933 -0.37416 -0.28032 -0.13211
0.8 -0.28924 -0.26069 -0.22359 -0.12284 -0.49315 -0.23385
: Numerical values of tangential skin friction co-efficient ( ) when
M\Kw 0 0.1 0.3 0.5 1 3
0 0.06425 0.06366 0.06308 0.06333 -0.00022 0.00000
0.1 0.06425 0.06365 0.06309 0.06330 0.05536 -40.33956
147
0.3 0.06421 0.06364 0.06312 0.06310 0.23411 0.00000
0.5 0.06415 0.06360 0.06320 0.06280 0.00003 0.04601
0.8 0.06401 0.06353 0.06345 0.06235 0.05010 0.05752
1 0.06391 0.06348 0.06380 0.06223 0.05415 0.06502
3 0.06329 0.06315 0.06294 0.06297 0.06073 0.05839
: Numerical values of local heat flux ( ) when
Nt\Nb 0.1 0.2 0.3 0.5 0.8 1
0.1 0.05710 0.05183 0.04690 0.03810 0.02753 0.02210
0.2 0.05389 0.04866 0.04380 0.03518 0.02499 0.01985
0.3 0.05059 0.04541 0.04061 0.03218 0.02238 0.01752
0.5 0.04361 0.03853 0.03386 0.02582 0.01678 0.01249
0.8 0.03179 0.02678 0.02225 0.01461 0.00644 0.00285
1.0 0.02235 0.01723 0.01260 0.00484 0.00340 0.00696
: Numerical values of local heat flux ( ) when
Pr\Sc 0.1 0.3 0.5 0.8 1 3
0.1 0.02546 0.02541 0.02539 0.02538 0.02538 0.02536
148
0.3 0.03143 0.03113 0.03108 0.03105 0.03104 0.03101
0.8 0.04449 0.04175 0.04113 0.04083 0.04073 0.04051
1.0 0.04779 0.04364 0.04260 0.04207 0.04191 0.04156
10.0 0.09057 0.02769 0.04147 0.16607 0.28697 0.09585
100.0 0.42154 0.41740 0.39706 0.47281 3.22413 0.45802
: Numerical values of local heat flux ( ) when
M\Kw 0 0.1 0.3 0.5 1 3
0 0.70720 0.69057 0.67582 0.71129 0.00026 0.00000
0.1 0.70704 0.69040 0.67657 0.71201 -0.02404 40.66124
0.3 0.70580 0.68911 0.68270 0.71785 -0.59007 0.00000
0.5 0.70343 0.68668 0.69581 0.72967 0.00001 0.73883
0.8 0.69827 0.68142 0.73369 0.75964 0.73477 0.76210
1 0.69412 0.67725 0.77690 0.78948 0.75472 0.77729
3 0.65383 0.63942 0.60378 0.66594 0.71312 0.71049
: Numerical values of local mass flux ( ) when
Nt\Nb 0.1 0.2 0.3 0.5 0.8 1.0
149
0.1 0.53927 0.54250 0.54344 0.54397 0.54392 0.54375
0.2 0.54152 0.54572 0.54684 0.54726 0.54681 0.54634
0.3 0.54804 0.55092 0.55146 0.55114 0.54989 0.54901
0.5 0.57416 0.56738 0.56435 0.56059 0.55664 0.55456
0.8 0.64866 0.60846 0.59369 0.57955 0.56850 0.56357
1.0 0.72735 0.64978 0.62210 0.59691 0.57869 0.57096
: Numerical values of local mass flux ( ) when
Pr\Sc 0.1 0.3 0.5 0.8 1 3
0.1 0.13529 0.25655 0.35086 0.47088 0.54216 1.08021
0.3 0.12994 0.25480 0.35007 0.47071 0.54221 1.08094
0.8 0.12532 0.25595 0.35286 0.47455 0.54642 1.08610
1.0 0.12377 0.25692 0.35460 0.47672 0.54873 1.08872
10.0 0.08717 0.31645 0.51361 0.78764 0.98737 0.97899
100.0 0.39419 0.31990 0.19249 0.49566 10.20110 0.16633
: Numerical values of local mass flux ( ) when
150
7.5 Conclusion
We have discussed the oblique MHD Walter-B nanofluid flow over a convective surface. The
governing differential equations of the problem are presented by means of suitable similarities
and then solved by Spectral Quasilinearization method (QLM) along with Spectral Local
linearization method (LLM). The graphical results for normal and tangential velocities,
temperature and concentration have been presented against several emerging non-dimensional
parameters. Numerical values of local surface shear stresses and heat as well as mass flux at the
wall are offered in tabulated form and discussed. Following conclusions are drawn from this
study:
Magnetic field parameter decreases the normal and tangential velocity profiles where
as it increases the fluid temperature.
Thermophoresis and Brownian motion has opposite influence on temperature and
concentration profiles.
Magnetic field parameter and local Weissenberg number decreases the local heat
and mass flux at the convective surface.
Prandtl number and Schmidt number have opposite influence on local heat and
mass flux.
151
Chapter 8
Effect of internal heat generation/absorption on an obliquely striking
Maxwell fluid past a convective surface
8.1 Introduction
This chapter numerically examines internal heat generation/absorption effects on oblique flow of
a Maxwell fluid with convective boundary condition. By transforming the governing equations
via suitable similarity transformations, Spectral Quasilinearization Method (QLM) along with
Spectral Local Linearization Method (LLM) is applied to solve the system of equations. The
velocity, temperature and local heat flux rate at the stretching convective surface are presented
through graphs against pertinent flow parameters and analysed. It is found that elasticity
parameter has opposite influence on normal and tangential components of velocity while it
increases the temperature profile. Moreover temperature profile rises with heat generation
(i.e. ). Local heat flux at the convective surface decreases with heat generation and it
increases with an increase in heat absorption parameter.
8.2 Problem Formulation
Consider steady, incompressible, two-dimensional, obliquely striking Maxwell fluid past a
stretching surface. It is presumed that the flow is being restrained in the region where is
the coordinate measured normal to the stretching surface. We further assume that the stretching
velocity is and the velocity far away is due to oblique flow and are
152
constants of inverse time dimension. Fig ( ) describes the non-dimensional model and the
coordinate system. The Cauchy stress tensor for Maxwell fluid is given by
Where is the zero shear rate viscosity, the relaxation time of Maxwell fluid,
(
) is the deformation rate tensor and
is upper convective derivative of a tensor and can be
obtained by
, in which is the velocity gradient.
The basic steady conservation of mass, momentum and thermal energy can be written
as , -.
Fig ( ): Non-dimensional description of the problem
( )
.
/
( )
.
/
( )
( ) ( )
153
Where are velocities along coordinate axis, the effective kinematic viscosity,
temperature, effective thermal diffusivity, is the dimensional heat generation/absorption
coefficient and is specific heat. the convective temperature of stretched surface, is heat
transfer coefficient and is ambient fluid temperature. Keeping these assumptions in mind, the
suitable conditions can be stated as , -.
( )
( )
In order to obtain the similarity solutions of Eqs ( ) ( ) with boundary conditions ( ),
the dimension less variables can be chosen as , -
√
√
√
√
√
( )
Substituting, Eq ( ) into Eqs ( ) ( ) our governing system takes the following non-
dimensional form
( )
.
/
( )
.
/
( )
( )
Where is the non-dimensional relaxation time,
is the Prandtl number and
is the heat source/sink parameter.
154
Introducing the stream function relations
( )
Using Eq ( ) in Eqs ( ) ( ), by eliminating the pressure we get
( )
( )
{
.
/
.
/
}
{
.
/
.
/
} ( )
( )
In order to transform Eqs ( ) and ( ) in to ordinary differential equations, following , -
( ) ( ) ( ) ( ) ( )
Substituting Eq ( ) in above, we achieve the following system of coupled non-linear
differential equations
( ) ( )
( ) ( )
( ) ( )
Along with the transformed boundary conditions
( ) ( ) ( )
( ) ( )
( ) ( ) ( ( )) ( ) ( )
where
√
is the Biot number. Since ( .
/ at , Where signifies the
boundary layer displacement constant. Substituting Eq ( ) in above, we get (
)
and
Making use of the transformation
155
( ) ( ) ( )
Equations ( ) ( ) along with their boundary conditions ( ), finally take the form as
( ) (
) ( )
( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ( )) ( ) ( )
8.3 Non-dimensional quantity of interest
We are interested here in local heat flux, which can be determined by
.
/
( )
In non-dimensional form, we can write
( ) ( )
8.4 Numerical Solution
The governing system of equations ( ) ( ) are non-linear in nature so their solutions
cannot be found exactly. We have employed a numerical scheme called Chebyshev spectral
collocation to approximate the solution of equations( ) ( ). Before applying the
spectral method, the equations are linearized using the quasi-linearization (QLM) approach that
was initially proposed by Bellman and Kalaba , - for solving functional equations. Considering
that the momentum equation for determining f(y) is decoupled from the rest of the equations in
the system, this equation is solved for ( ) first using the QLM and spectral collocation method.
156
The approximate solution for ( ) is then substituted in the other two equations which are
solved, in turn, for ( ) and ( ). Applying the QLM on the system of equations give
( )
( )
( )
Subject to
( ) ( ) ( ) ( ( ))
( )
( ) ( ) ( ) ( )
Where
(
)
Equations ( ) ( ) constitute a sequence of linear and decoupled equations so they are
solved iteratively starting from a given initial approximation at . In this work, the
Chebyshev spectral collocation method approach as described in , - is chosen to
integrate ( ) – ( ). The domain in is transformed in to the interval , - so that the
spectral collocation method can be applied. For the convenience of the numerical computations,
157
the semi-infinite domain in is approximated by the truncated domain , - where is a finite
number selected to be large enough to represent the behaviour of the flow properties when is
very large. We use the transformation ( ) to map the interval , - to , - The
basic idea behind the spectral collocation method is the introduction of a differentiation matrix
which is used to approximate unknown derivatives of the function ( ) at the grid points by the
matrix vector product
( )
|
∑ ( ) ( )
where represents the total grid points,
and
, ( ) ( ) ( )-
denote vector function at collocation points. Similar vector functions corresponding to the other
unknown functions such as and are denoted by and respectively. Moreover, other higher
derivatives can be achieved by powers of as
( ) ( ) ( ) ( ) ( ) ( ) ( )
where is the order of the derivative. The matrix has the size ( ) ( ) The grid
points on are defined as
( )
Thus, applying the spectral method on equations ( ) ( ) give
( )
158
( )
( )
Where ( ) are ( ) ( ) matrices defined as:
( )
( )
( )
and ( ) are ( ) vectors obtained by evaluating the right hand sides of
( ) ( ) at the collocation points and incorporating the boundary conditions. The
approximate solutions for ( ) ( ) and ( ) are obtained by solving
equations( ) – ( ). Figs ( ) – ( ) are plotted to exhibit the residual errors for
velocities and temperature profiles against elasticity parameter
159
Fig ( ): Residual Error for Normal flow
( )
Fig ( ): Residual Error for tangential flow
( )
Fig ( ): Residual Error for temperature profile ( )
8.5 Analysis of Graphical Results
The main purpose of this section is to study the influence of emerging parameters on non-
dimensional quantities such as normal and tangential velocities, temperature profile and local
heat flux rate at the convective surface. Figs ( ) – ( ) are prepared for this purpose. Figs
160
( ) and ( ) describe the effects of Deborah number on normal ( ) and tangential
component of velocity ( ) respectively. It is quite obvious from these figures that Deborah
number has opposite influence on both the velocities. Normal component of the velocity
( ) decreases while tangential component of the velocity ( ) increases near the wall with an
increase in . The effects of Deborah number and Biot number on temperature profile
( ) are presented through Figs ( ) and ( ) and found that ( ) increases with Deborah
number and Biot number . Impact of heat source/sink parameter on the temperature profile
( ) is expressed through Figs ( ) and ( ). These results clearly indicate that temperature
profile ( ) increases with an increase in heat generation ( ) but it decreases with heat
absorption ( ) This is in good agreement with the physical situation because an internal
heating naturally causes the temperature of the fluid to rise up and vice versa. Fig ( ) shows
that temperature profile ( ) with prandtl number as higher Prandtl nubmer means less
thermal conductivity which casues the temperature to drop down. From Figs ( ) and ( ),
it is noticed that local heat flux rate ( ) increases with heat absorption parameter ( )
and it decreases with heat generation parameter ( ) Moreover, it is also observed from
these figures that local heat flux rate ( ) tends to decrease with an increase in Deborah
number for heat generation as well as absorption case. Figs ( ) and ( ) are plotted to
discover the effect of prandtl number Pr on local heat flux rate ( ) for heat source and sink
respectively. These results show that heat flux rate ( ) increase as we increase the values of
. This increasing behaviour is found to be quite similar for both heat genaration and absorption
cases. Finally, the influence of Biot number on the heat flux rate ( ) is discovered
through Fig ( ). It can be seen that heat flux rate ( ) increases with an increase in Biot
number .
161
Fig ( ): Velocity profile ( ) against
Fig ( ): Velocity profile ( ) against
162
Fig ( ): Temperature profile ( ) against
when
Fig ( ): Temperature profile ( ) against
when
163
Fig ( ): Temperature profile ( ) against heat generation
parameter ( ) when
Fig ( ): Temperature profile ( ) against heat absorption
164
parameter ( ) when
Fig ( ): Temperature profile ( ) against
when
Fig ( ): Local heat flux ( ) against for heat absorption
165
case ( ) when
Fig ( ): Local heat flux ( ) against for heat generation
case ( ) when
166
Fig ( ): Local heat flux ( ) against for heat absorption
case ( ) when
Fig ( ): Local heat flux ( ) against for heat generation
case ( ) when .
167
Fig ( ): Local heat flux ( ) against for heat generation
case ( )when .
8.6 Concluding Remarks
The oblique stagnation point Maxwell fluid flow has been explored numerically with internal
heat generation/absorption over a convective surface. The governing non-linear coupled ordinary
differential equations are solved through Spectral Quasilinearization Method (QLM) and
Spectral Local Linearization Method (LLM). The key points of this study are:
It is found that elasticity parameter has opposite influence on normal and tangential
components of velocity while it increases the temperature profile.
The Prandtl number enhances the surface heat transfer for heat generation as well as heat
absorption case.
Local heat flux decreases with heat generation and increases with an increase in heat
absorption parameter.
168
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